Closed form optimized transmission conditions for complex diffusion with many subdomains

Optimized transmission conditions in domain decomposition methods have been the focus of intensive research efforts over the past decade. Traditionally, transmission conditions are optimized for two subdomain model configurations, and then used in practice for many subdomains. We optimize here transmission conditions for the first time directly for many subdomains for a class of complex diffusion problems. Our asymptotic analysis leads to closed form optimized transmission conditions for many subdomains, and shows that the asymptotic best choice in the mesh size only differs from the two subdomain best choice in the constants, for which we derive the dependence on the number of subdomains explicitly, including the limiting case of an infinite number of subdomains, leading to new insight into scalability. Our results include both Robin and Ventcell transmission conditions, and we also optimize for the first time a two-sided Ventcell condition. We illustrate our results with numerical experiments, both for situations covered by our analysis and situations that go beyond.